1
Log into your Microsoft account, in there you should be able to remove your old PC using the steps on this link, but do not do that if you need to move a Windows licence to the new PC.
However, many people are reporting at the moment, that when they delete a device from their account online, the device keeps reappearing on the account, it seems to be a server side problem that Microsoft needs to fix.
2
To add a new device to your Microsoft account, if that is not a Microsoft manufactured device like a Surface laptop, you cannot use the serial number to add the device.
Ensure you log into Windows with your Microsoft account on the new device, then enable the Location Services setting in the Settings app and also the 'Find my Device setting, your new device will then automatically get associated to your Microsoft account, that may not happen immediately, but it will happen.